Theorem wunss

Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	wunss.3	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion	wunss	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		wunss.3	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
4	1 2	wunpw	⊢ ( 𝜑 → 𝒫 𝐴 ∈ 𝑈 )
5	1 4	wunelss	⊢ ( 𝜑 → 𝒫 𝐴 ⊆ 𝑈 )
6	2 3	sselpwd	⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐴 )
7	5 6	sseldd	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )